Gauge Freedom and Metric Dependence in Neural Representation Spaces

TL;DR

This paper explores the geometric gauge freedom in neural representations, showing that common similarity measures are coordinate-dependent and proposing invariant analysis methods.

Contribution

It introduces a geometric framework for neural representations considering gauge freedom, explaining instability of similarity measures and suggesting invariant analysis approaches.

Findings

Invertible transformations can distort similarity measures without changing model predictions.

Cosine similarity and nearest-neighbor structures are sensitive to coordinate changes.

Invariant analysis methods are necessary for meaningful neural representation comparisons.

Abstract

Neural network representations are often analyzed as vectors in a fixed Euclidean space. However, their coordinates are not uniquely defined. If a hidden representation is transformed by an invertible linear map, the network function can be preserved by applying the inverse transformation to downstream weights. Representations are therefore defined only up to invertible linear transformations. We study neural representation spaces from this geometric viewpoint and treat them as vector spaces with a gauge freedom under the general linear group. Within this framework, commonly used similarity measures such as cosine similarity become metric-dependent quantities whose values can change under coordinate transformations that leave the model function unchanged. This provides a common interpretation for several observations in the literature, including cosine-similarity instability, anisotropy in embedding spaces, and the appeal of representation comparison methods such as SVCCA and CKA. Experiments on multilayer perceptrons and convolutional networks confirm that inserting invertible transformations into trained models can substantially distort cosine similarity and nearest-neighbor structure while leaving predictions unchanged. These results indicate that analysis of neural representations should focus either on quantities that are invariant under this gauge freedom or on explicitly chosen canonical coordinates.